The motivation behind the class of coupled systems that we study comes from biological predator-prey models and other population and social dynamics phenomena in their linearized versions (see [1]}). Also, in chemical processes (see [2]}), coupled temperature-concentration parabolic PDEs occur to describe the process dynamics. Boundary stabilization of parabolic systems with same number of controls as the number of equations has been achieved via backstepping, see [3]} and via modal decomposition in [4]}. For the case of underactuated systems, boundary stabilization has been achieved via backstepping for some classes of hyperbolic systems, see [5]}, [6]}, [7]}, however for the class of parabolic systems, strict assumptions should be imposed on the internal dynamics, see [8]}. In the latter, boundary stabilization was achieved only for \(2\times 2\)  parabolic systems if the second equation is asymptotically stable up to the coupling term. The difficulty in achieving boundary stabilization for general classes of underactuated systems is revealed in controllability studies, see [9]}. In such works, it is seen that approximate controllability leading to boundary stabilization is only possible under specific assumptions, such as when all diffusion coefficients are identical. For the general case of \(m\)  equations with less than \(m\)  controlled ones and in the presence of distinct diffusion coefficients, only internal controllability is possible leading to internal stabilization. Thus, to deal with the general case, we can apply stabilizing controls internally. Various studies on internal controllability of underactuated systems (see [10]}, [11]}, [12]}, [13]}) reveal that the difficulty grows with the number of the states and the number of distinct diffusion coefficients as a result of the notion of algebraic solvability (see [14]}). The problem of stabilization of such systems runs deep, see for instance [15]}, [16]}, [17]}, [18]}, see also the seminal work of [19]} on distributed control. However, to the best of our knowledge, explicit stabilizing laws for these underactuated systems have not been determined yet.
